Advances in High Energy Physics
Volume 2018 (2018), Article ID 2741694, 10 pages
https://doi.org/10.1155/2018/2741694

Research Article

Generalized Dirac Oscillator in Cosmic String Space-Time

Lin-Fang Deng,¹ Chao-Yun Long,1,2 Zheng-Wen Long,¹ and Ting Xu¹

¹Department of Physics, Guizhou University, Guiyang 550025, China
²Laboratory for Photoelectric Technology and Application, Guizhou University, Guiyang 550025, China

Correspondence should be addressed to Chao-Yun Long; longchaoyun66@163.com

Received 15 July 2018; Revised 6 September 2018; Accepted 9 October 2018; Published 18 November 2018

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2018 Lin-Fang Deng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

Abstract.

In this work, the generalized Dirac oscillator in cosmic string space-time is studied by replacing the momentum with its alternative In particular, the quantum dynamics is considered for the function to be taken as Cornell potential, exponential-type potential, and singular potential. For Cornell potential and exponential-type potential, the corresponding radial equations can be mapped into the confluent hypergeometric equation and hypergeometric equation separately. The corresponding eigenfunctions can be represented as confluent hypergeometric function and hypergeometric function. The equations satisfied by the exact energy spectrum have been found. For singular potential, the wave function and energy eigenvalue are given exactly by power series method.

1. Introduction

In quantum mechanics, there has been an increasing interest in finding the analytical solutions that play an important role for getting complete information about quantum mechanical systems [1–3]. The Dirac oscillator proposed in [4] is one of the important issues in this relativistic quantum mechanics recently. In this quantum model, the coupling proposed is introduced in such a way that the Dirac equation remains linear in both spatial coordinates and momenta and recovers the Schrödinger equation for a harmonic oscillator in the nonrelativistic limit of the Dirac equation [4–11]. As a solvable model of relativistic quantum mechanical system, the Dirac oscillator has many applications and has been studied extensively in different field such as high-energy physics [12–15], condensed matter physics [16–18], quantum Optics [19–25], and mathematical physics [26–33]. On the other hand, the analysis of gravitational interactions with a quantum mechanical system has recently attracted a great deal attention and has been an active field of research [5, 6, 34–43]. The study of quantum mechanical problems in curved space-time can be considered as a new kind of interaction between quantum matter and gravitation in the microparticle world. In recent years, the Dirac oscillator embedded in a cosmic string background has inspired a great deal of research such as the dynamics of Dirac oscillator in the space-time of cosmic string [44–47], Aharonov-Casher effect on the Dirac oscillator [5, 48], and noninertial effects on the Dirac oscillator in the cosmic string space-time [49–51]. It is worth mentioning that based on the coupling corresponding to the Dirac oscillator a new coupling into Dirac equation first has been proposed by Bakke et al. [52] and used in different fields [53–57]. This model is called the generalized Dirac oscillator which in special case is reduced to ordinary Dirac oscillator. Inspired by the above work, the main aim of this paper is to analyze the generalized Dirac oscillator model with the interaction functions taken as Cornell potential, singular potential and exponential-type potential in cosmic string space-time and to find the corresponding energy spectrum and wave functions. This work is organized as follows. In Section 2, the new coupling is introduced in such a way that the Dirac equation remains linear in momenta, but not in spatial coordinates in a curved space-time. In Sections 3, 4, and 5, we concentrate our efforts in analytically solving the quantum systems with different function and find the corresponding energy spectrum and spinors, respectively. In Section 6, we make a short conclusion.

2. Generalized Dirac Oscillator with a Topological Defect

In cosmic string space-time, the general form of the cosmic string metric in cylindrical coordinates read [41, 42, 44, 58, 59]

with , , and . The parameter is related to the linear mass density of string by and runs in the interval . In the limit as we get the line element of cylindrical coordinates. The Dirac equation in the curved space-time reads

where the matrices are the generalized Dirac matrices defining the covariant Clifford algebra , m is mass of the particle, and is the spinor affine connection. We choose the basis tetrad as

then in this representation the matrices [44] can be found to be

It is well known that, in both Minkowski space-time and curved space-time, usual Dirac oscillator can be obtained by the carrying out nonminimal substitution in Dirac equation where and are the mass and oscillator frequency. In the following, we will construct the generalized oscillator in curved space-time. To do this end, we can replace momenta in the Dirac equation of curved space-time by

where are undetermined functions of . It is to say that we introduce a new coupling in such a way that the Dirac equation remains linear in momenta but not in coordinates. In particular, in this work, we only consider the radial component the nonminimal substitution

By introducing this new coupling (6) into (2) and with the help of (4), in cosmic string space-time the eigenvalue equation of generalized Dirac oscillator can be written as

We choose the following ansatz:

then we have

where

It is straightforward to prove the following relations satisfied above matrices :

With help of (12) and simple algebraic calculus, (9) becomes

It is easy to prove the following relation [44]:

where . The eigenvalue of can be assumed as and (13) reads

where

For the component , from (10) an analog equation can be also obtained

where

In particular, and will be reduced to the result obtained in [44] when the function is taken as . As we can see, and have the same form. So without loss of generality in remaining parts of this work, our main tasks is only to solve the equation with different functions and find corresponding eigenvalue and eigenfunction. While with regard to , it is straightforward to obtain the corresponding solution.

3. The Solution with (ρ) to Be Cornell Potential

The Cornell potential that consists of Coulomb potential and linear potential has gotten a great deal of attention in particle physics and was used with considerable success in models describing systems of bound heavy quarks [60–62]. In Cornell potential, the short-distance Coulombic interaction arises from the one-gluon exchange between the quark and its antiquark, and the long-distance interaction is included to take into account confining phenomena.

Now we let the function be Cornell potential

where and are two constants. Substituting (17) into and leads to following equation:

where

We make a change in variables and then (18) can be rewritten as

Taking account of the boundary conditions satisfied by the wave function , i.e., for and for , physical solutions can be expressed as [44, 60, 63, 64]

If we insert this wave function into (20), we have the second-order homogeneous linear differential equation in the following form:

It is well known that (22) is the confluent hypergeometric equation and it is immediate to obtain the corresponding eigenvalues and eigenfunctions

with

In particular, if we assume that , from (24), the energy levels of generalized Dirac oscillator with to be Cornell potential in the absence of a topological defect can be obtained. In addition if we let , in (24) the energy levels given here will be reduced to that one obtained in [44].

4. The Solution with (ρ) to Be Singular Potential

The investigation of singular potentials in quantum mechanics is almost as old as quantum mechanics itself and covers a wide range of physical and mathematical interest because the real world interactions were likely to be highly singular [65]. The singular potentials of type, with , are of great current physical interest and are relevant to many problems such as the three-body problem in nuclear physics [66, 67], point-dipole interactions in molecular physics [68], the tensor force between nucleons [69], and the interaction between a charges and an induced dipole [70], respectively. Recently, in cosmic string background, singular inverse-square potential with a minimal length had been studied [71].

Next let us take to be singular inverse-square-type potential [72]

Substituting (26) into , the corresponding radial equation reads

where

It is obvious that (27) has the same mathematical structure with the Schrödinger equation of fourth-order inverse-potential in [73]. So (27) can be solved by power series method.

We look for an exact solution of (27) via the following ansatz to the radial wave function [73–75]:

Thence, (27) can be transformed into the following form:

Now we take in following series form:

Substituting (31) into (30) gives rise to following equation:

To make (32) be valid for all values of , the coefficients of each term of the polynomial of must be equal to zero separately. We, therefore, obtain

Using , (33a), and (33e) and after simple algebraic calculation, the corresponding energy can be written as

The general radial wave functions corresponding to the energy spectra given in (34) are

With the help of (32) and (33a)–(33e), the expansion coefficients in (35) satisfy the following recursion relation [73]:

From the recursion relation (36) we can determine the coefficients of the power series in terms of and . In addition the above recursion relation implies that (35) yields one solution as a power series in even powers of and another in odd powers of .

In addition, (27) can be also mapped to the double-confluent Heun equation by appropriate function transformation [76]. So when is taken as singular inverse-square-type potential, the solutions of (27) can be also given by the solution of the double-confluent Heun equation [76, 77].

5. The Solution with (ρ) to Be Exponential-Type Potential

The exponential-type potentials are very important in the study of various physical systems, particularly for modeling diatomic molecules. The typical exponential-type potentials include Eckart potentials [78], the Morse potential [79, 80], the Wood–Saxon potential [81], and Hulthén potential [82, 83]. The research work on the Dirac equation with the above potential is mainly concentrated on Minkowski time and space. However, it has been noticed recently that it is also interesting to study this kind quantum systems in a cosmic string background [84]. In this section we will take the as exponential-type function and solve the corresponding Dirac equation in cosmic string space-time.

As is known to all, the Dirac equation and Schrödinger equation have been studied by resorting different methods. A usual way is transforming the eigenvalue equation of quantum system considered into a solvable equation via suitable variable substitutions and function transformations [85–87]. In order to obtain solution for being exponential-type potential, we firstly consider the following linear second-order differential equation

where are constants. It is known that singular points of a differential equation determine the form of solutions. In this equation, there are two singular points, i.e., x = 0 and x = 1. In order to remove these singularities and get physically acceptable solutions we use the following ansatz:

where and are two real parameters. Further we make this two parameters to satisfy following relationships:

and by substituting (38) into (37), the differential equation for can be written as

with . In other words, (37) is reduced to the well-known hypergeometric equation, when condition (39) is imposed. Making use of the boundary conditions at and [87, 88], we can find the equation obeyed by the energy eigenvalue:

and the corresponding eigenfunctions is given in terms of the Gauss hypergeometric functions below

where A is normalization constant. Next, we will use the results given here to obtain the solutions of Dirac equation exponential-type interaction in cosmic string space-time. As a direct application of the above method, let us take the function to be as Yukawa potential, Hulthén-Type potential, and generalized Morse potential, respectively.

Case 1 ( being Yukawa potential). In Yukawa meson theory, the Yukawa potential firstly was introduced to describe the interactions between nucleons [89]. Afterwards, it has been applied to many different areas of physics such as high-energy physics [90, 91], molecular physics [92], and plasma physics [93]. In recent years, the considerable efforts have also been made to study the bound state solutions by using different methods.
Now let us choose to be Yukawa potential

then takes the form

However, the radial equation (44) cannot accept exact solution due to the presence of the centrifugal term [86]. In order to find analytical solution, we have to use some approximation approaches for the centrifugal term potential. Following [87], the approximation for the centrifugal term reads

It is worth mentioning that the above approximations are valid for [87]. So if we make the control parameter small enough, then we can guarantee that the above approximations in (45) hold for larger values . In other words, this approximation (45) is valid in our work.
Using the approximation in (45) and setting

in the new function and new variable , (44) becomes

where

Comparing (47) with (37) and using the results given in (41) and (42), it is not difficult to find the equation obeyed by eigenvalues and eigenfunctions and they can be given, respectively,

where

Case 2 ( being Hulthén-type potential). In this section, we are interested in considering the Hulthén potential that describes the interaction between two atoms and has been used in different areas of physics and attracted a great of interest for some decades [82, 83, 94]. Next we take the interaction function being Hulthén-Type potential

where , , and are real constants. Inserting (45) and (51) into , then can written as

In the same way as in previous section, taking into consideration approximation (45) for the centrifugal term and using the variable transformation and function transformation , (51) changes

where

With the help of (38), (41), and (42), the solutions for being Hulthén-Type potential can be easily obtained and the corresponding eigenvalues and eigenfunctions are, respectively,

where

Case 3 ( being generalized Morse potential). The Morse potential [79, 80] as an important molecular potential describes the interaction between two atoms. We choose the interaction function being generalized Morse potential

As before, substitution of form (59) into and straightforward calculation lead to the following equation:

Letting and , the above differential equation (59) changes into the form

where

It is easy to see that the differential equation (60) is also similar to (37). So again according to the quantization condition (40) the corresponding expression of eigenvalues can be written as

The wave function in this case read

where

The above results show that the radial equation of the generalized Dirac oscillator with interaction function to be taken as the exponential-type potential can be mapped into the well-known hypergeometric equation and the analytical solutions can have been found.

6. Conclusion

In this work, the generalized Dirac oscillator has been studied in the presence of the gravitational fields of a cosmic string. The corresponding radial equation of generalized Dirac oscillator is obtained. In our generalized Dirac oscillator model, we take the interaction function to be as Cornell potential, Yukawa potential, generalized Morse potential, Hulthén-Type potential, and singular potential, respectively. By solving the corresponding wave equations the corresponding energy eigenvalues and the wave functions have been obtained and we have showed how the cosmic string leads to modifications in the spectrum and wave function. Based on consideration that Dirac oscillator has been studied extensively in high-energy physics, condensed matter physics, quantum optics, mathematical physics, and even connection with Higgs symmetry it also makes sense to generalize the generalized Dirac oscillator to these fields.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11565009).

References

C. Quigg and J. L. Rosner, “Quantum mechanics with applications to quarkonium,” Physics Reports, vol. 56, no. 4, pp. 167–235, 1979.

H. Hassanabadi, S. Zarrinkamar, and A. A. Rajabi, “A simple efficient methodology for Dirac equation in minimal length quantum mechanics,” Physics Letters B, vol. 718, no. 3, pp. 1111–1113, 2013.

E. Hackmann, B. Hartmann, C. Lämmerzahl, and P. Sirimachan, “Test particle motion in the space-time of a Kerr black hole pierced by a cosmic string,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 82, no. 4, 2010.

M. Moshinsky and A. Szczepaniak, “The Dirac oscillator,” Journal of Physics A: Mathematical and General, vol. 22, no. 17, pp. L817–L819, 1989.

K. Bakke and C. Furtado, “On the interaction of the Dirac oscillator with the Aharonov–Casher system in topological defect backgrounds,” Annals of Physics, vol. 336, pp. 489–504, 2013.

K. Bakke, “Rotating effects on the Dirac oscillator in the cosmic string spacetime,” General Relativity and Gravitation, vol. 45, no. 10, pp. 1847–1859, 2013.

D. Itô, K. Mori, and E. Carriere, “An example of dynamical systems with linear trajectory,” Nuovo Cimento A, vol. 51, p. 1119, 1967.

H. Hassanabadi, Z. Molaee, and S. Zarrinkamar, “DKP oscillator in the presence of magnetic field in (1+2)-dimensions for spin-zero and spin-one particles in noncommutative phase space,” The European Physical Journal C, vol. 72, article 2217, 2012.

O. Aouadi, Y. Chargui, and M. S. Fayache, “The Dirac oscillator in the presence of a chain of delta-function potentials,” Journal of Mathematical Physics, vol. 57, no. 2, 023522, 10 pages, 2016.

A. Albrecht, A. Retzker, and M. B. Plenio, “Testing quantum gravity by nanodiamond interferometry with nitrogen-vacancy centers,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 90, no. 3, Article ID 033834, 2014.

M. Presilla, O. Panella, and P. Roy, “Quantum phase transitions of the Dirac oscillator in the anti-Snyder model,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 92, no. 4, 2015.

F. M. Andrade, E. O. Silva, Ferreira. M. M., and E. C. Rodrigues, “On the κ-Dirac Oscillator revisited,” Physics Letters B, vol. 731, no. 15, pp. 327–330, 2014.

H. Hassanabadi, S. S. Hosseini, A. Boumali, and S. Zarrinkamar, “The statistical properties of Klein-Gordon oscillator in noncommutative space,” Journal of Mathematical Physics, vol. 55, no. 3, Article ID 033502, 2014.

J. Munarriz, F Dominguez-Adame, and R. P. A. Lima, “Spectroscopy of the Dirac oscillator perturbed by a surface delta potential,” Physics Letters A, vol. 376, no. 46, pp. 3475–3478, 2012.

J. Grineviciute and D. Halderson, “Relativistic,” Physical Review C: Nuclear Physics, vol. 85, no. 5, Article ID 054617, 2012.

A. Faessler, V. I. Kukulin, and M. A. Shikhalev, “Description of intermediate- and short-range NN nuclear force within a covariant effective field theory,” Annals of Physics, vol. 320, no. 1, pp. 71–107, 2005.

V. M. Villalba, “Exact solution of the two-dimensional Dirac oscillator,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 49, no. 1, 586 pages, 1994.

C. Quesne and V. M. Tkachuk, “Dirac oscillator with nonzero minimal uncertainty in position,” Journal of Physics A: Mathematical and General, vol. 38, no. 8, pp. 1747–1765, 2005.

P. L. Knight, “Quantum Fluctuations and Squeezing in the Interaction of an Atom with a Single Field Mode,” Physica Scripta, vol. 12, pp. 51–55, 1986.

P. Rozmej and R. Arvieu, “The Dirac oscillator. A relativistic version of the Jaynes-Cummings model,” Journal of Physics A: Mathematical and General, vol. 32, no. 28, pp. 5367–5382, 1999.

F. M. Toyama, Y. Nogami, and F. A. Coutinho, “Behaviour of wavepackets of the "DIRac oscillator'': DIRac representation versus Foldy-Wouthuysen representation,” Journal of Physics A: Mathematical and General, vol. 30, no. 7, pp. 2585–2595, 1997 (Arabic).

A. Bermudez, M. A. Martin-Delgado, and E. Solano, “Exact mapping of the 2 + 1 Dirac oscillator onto the Jaynes-Cummings model: Ion-trap experimental proposal,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 76, no. 4, Article ID 041801, 2007.

J. M. Torres, E. Sadurni, and T. H. Seligman, “Two interacting atoms in a cavity: exact solutions, entanglement and decoherence,” Journal of Physics A: Mathematical and General, vol. 43, no. 19, 192002, 8 pages, 2010.

A. Bermudez, M. A. Martin-Delgado, and E. Solano, “Mesoscopic Superposition States in Relativistic Landau Levels,” Physical Review Letters, vol. 99, Article ID 123602, 2007.

L. Lamata, J. León, T. Schätz, and E. Solano, “Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion,” Physical Review Letters, vol. 98, no. 25, 2007.

M. Hamzavi, M. Eshghi, and S. M. Ikhdair, “Effect of tensor interaction in the Dirac-attractive radial problem under pseudospin symmetry limit,” Journal of Mathematical Physics, vol. 53, no. 082101, 2012.

C.-K. Lu and I. F. Herbut, “Supersymmetric Runge-Lenz-Pauli vector for Dirac vortex in topological insulators and graphene,” Journal of Physics A: Mathematical and Theoretical, vol. 44, no. 29, Article ID 295003, 2011.

S. Zarrinkamar, A. A. Rajabi, and H. Hassanabadi, “Dirac equation in the presence of coulomb and linear terms in (1+ 1) dimensions; the supersymmetric approach,” Annals of Physics, vol. 325, pp. 1720–1726, 2010.

Y. Chargui, A. Trabelsi, and L. Chetouani, “Bound-states of the (1+1)-dimensional DKP equation with a pseudoscalar linear plus Coulomb-like potential,” Physics Letters A, vol. 374, no. 29, pp. 2907–2913, 2010.

A. Bermudez, M. A. Martin-Delgado, and A. Luis, “Chirality quantum phase transition in the Dirac oscillator,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 77, no. 6, 2008.

D. A. Kulikov, R. S. Tutik, and A. P. Yaroshenko, “Relativistic two-body equation based on the extension of the SL(2,C) group,” Modern Physics Letters B, vol. 644, no. 5-6, pp. 311–314, 2007.

A. S. de Castro, P. Alberto, R. Lisboa, and M. Malheiro, “Relating pseudospin and spin symmetries through charge conjugation and chiral transformations: The case of the relativistic harmonic oscillator,” Physical Review C, vol. 73, Article ID 054309, 2006.

J. A. Franco-Villafañe, E. Sadurní, S. Barkhofen, U. Kuhl, F. Mortessagne, and T. H. Seligman, “First Experimental Realization of the Dirac Oscillator,” Physical Review Letters, vol. 111, Article ID 170405, 2013.

L. C. N. Santos and C. C. Barros, “Scalar bosons under the influence of noninertial effects in the cosmic string spacetime,” Eur. Phys. J. C, vol. 77, no. 186, 2017.

K. Bakke and C Furtado, “Anandan quantum phase for a neutral particle with Fermi–Walker reference frame in the cosmic string background,” The European Physical Journal C, vol. 69, pp. 531–539, 2010.

M. Hosseinpour, F. M. Andrade, E. O. Silva, and H. Hassanabadi, “Erratum to: Scattering and bound states for the Hulthén potential in a cosmic string background,” The European Physical Journal C, vol. 77, no. 270, 2017.

M. Falek, M. Merad, and T. Birkandan, “Duffin-Kemmer-Petiau oscillator with Snyder-de SITter algebra,” Journal of Mathematical Physics, vol. 58, no. 2, 023501, 13 pages, 2017.

K. Bakke and J. Plus, “Noninertial effects on the Dirac oscillator in a topological defect spacetime,” Eur. Phys. J. Plus, vol. 127, no. 82, 2012.

F. M. Andrade and E. O. Silva, “Effects of spin on the dynamics of the 2D Dirac oscillator in the magnetic cosmic string background,” European Physical Journal C, vol. 74, no. 3187, 2014.

E. R. Figueiredo Medeiros and E. R. Bezerra de Mello, “Relativistic quantum dynamics of a charged particle in cosmic string spacetime in the presence of magnetic field and scalar potential,” The European Physical Journal C, vol. 72, article 2051, 2012.

L. B. Castro, “Quantum dynamics of scalar bosons in a cosmic string background,” The European Physical Journal C, vol. 75, no. 287, 2015.

K. Jusufi, “Noninertial effects on the quantum dynamics of scalar bosons,” The European Physical Journal C, vol. 76, no. 61, 2016.

G. A. Marques, V. B. Bezerra, and S. G. Fernandes, “Exact solution of the dirac equation for a coulomb and scalar poten-tials in the gravitational field of a cosmic string,” Physics Letters A, vol. 341, no. 1-4, pp. 39–47, 2005.

J. Carvalho, C. Furtado, and F. Moraes, “Dirac oscillator interacting with a topological defect,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 84, no. 3, Article ID 032109, 2011.

D. Chowdhury and B. Basu, “Effect of a cosmic string on spin dynamics,” Physical Review D, vol. 90, Article ID 125014, 2014.

H. Hassanabadi, A. Afshardoost, and S. Zarrinkamar, “On the motion of a quantum particle in the spinning cosmic string space-time,” Annals of Physics, vol. 356, pp. 346–351, 2015.

N. Ferkous and A. Bounames, “Energy spectrum of a 2D Dirac oscillator in the presence of the Aharonov—Bohm effect,” Physics Letters A, vol. 325, no. 1, pp. 21–29, 2004.

H. F. Mota and K. Bakke, “Noninertial effects on the ground state energy of a massive scalar field in the cosmic string spacetime,” Physical Review D, vol. 89, Article ID 027702, 2014.

K. Bakke, “Torsion and noninertial effects on a nonrelativistic Dirac particle,” Annals of Physics, vol. 346, pp. 51–58, 2014.

P. Strange and L. H. Ryder, “The Dirac oscillator in a rotating frame of reference,” Physics Letters A, vol. 380, no. 42, pp. 3465–3468, 2016.

J. Anandan, “Gravitational and rotational effects in quantum interference,” Physical Review D, vol. 15, no. 1448, 1977.

K. Bakke and C. Furtado, “On the confinement of a Dirac particle to a two-dimensional ring,” Physics Letters A, vol. 376, no. 15, pp. 1269–1273, 2012.

M. J. Bueno, J. Lemos de Melo, C. Furtado, and A. M. de M. Carvalho, “Quantum dot in a graphene layer with topological defects,” The European Physical Journal Plus, vol. 129, no. 9, 2014.

J. Amaro Neto, M. J. Bueno, and C. Furtado, “Two-dimensional quantum ring in a graphene layer in the presence of a Aharonov-Bohm flux,” Annals of Physics, vol. 373, pp. 273–285, 2016.

Neto. Jose Amaro, J. R. de S, Claudio. Furtadoa, and Sergei. Sergeenkov, “Quantum ring in gapped graphene layer with wedge disclination in the presence of a uniform magnetic field,” European Physical Journal Plus 133, vol. 185, 2018.

D. Dutta, O. Panella, and P. Roy, “Pseudo-Hermitian generalized Dirac oscillators,” Annals of Physics, vol. 331, pp. 120–126, 2013.

H. P. Laba and V. M. Tkachuk, 2018, https://arxiv.org/pdf/1804.06091.pdf.

E. R. Bezerra de Mello and A. A. Saharian, “Fermionic current induced by magnetic flux in compactified cosmic string spacetime,” The European Physical Journal C, vol. 73, no. 8, 2013.

S. Bellucci, E. R. Bezerra de Mello, A. de Padua, and A. A. Saharian, “Fermionic vacuum polarization in compactified cosmic string spacetime,” The European Physical Journal C, vol. 74, no. 1, 2014.

J. Domenech-Garret and M. Sanchis-Lozano, “Spectroscopy, leptonic decays and the nature of heavy quarkonia,” Physics Letters B, vol. 669, no. 1, pp. 52–57, 2008.

E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, and T. -. Yan, “Erratum: Charmonium: The model,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 21, no. 1, pp. 313–313, 1980.

E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, and T. M. Yan, “Charmonium: comparison with experiment,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 21, no. 1, pp. 203–233, 1980.

J. Carvalho, A. M. M. Carvalho, E. Cavalcante, and C. Furtado, “Klein–Gordon oscillator in Kaluza–Klein theory,” The European Physical Journal C, vol. 76, no. 7, p. 365, 2016.

F. Ahmed, “The energy–momentum distributions and relativistic quantum effects on scalar and spin-half particles in a Gödel-type space–time,” The European Physical Journal C, vol. 78, no. 7, p. 598, 2018.

C. P. Burgess, P. Hayman, M. Rummel, M. Williams, and L. Zalavári, “Point-Particle Effective Field Theory III: Relativistic Fermions and the Dirac Equation,” Journal of High Energy Physics, vol. 04, no. 106, 2017.

M. Bawin and S. A. Coon, “Singular inverse square potential, limit cycles, and self-adjoint extensions,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 67, no. 4, 2003.

S. R. Beane, P. F. Bedaque, L. Childress, A. Kryjevski, J. McGuire, and U. van Kolck, “Singular potentials and limit cycles,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 64, no. 4, 2001.

J. Lévy-Leblond, “Electron Capture by Polar Molecules,” Physical Review Journals Archive, vol. 153, no. 1, pp. 1–4, 1967.

U. van Kolck and Prog. Part, “Prog. Part. Nucl. Phys. 43, 409, 1999”.

E. Vogt and G. H. Wannier, “Scattering of ions by polarization forces,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 95, no. 5, pp. 1190–1198, 1954.

D. Bouaziz and M. Bawin, “Singular inverse square potential in arbitrary dimensions with a minimal length: Application to the motion of a dipole in a cosmic string background,” Physical Review A, vol. 78, 2010.

D. Bouaziz and M. Bawin, “Singular inverse-square potential: Renormalization and self-adjoint extensions for medium to weak coupling,” Physical Review A, vol. 89, Article ID 022113, 2014.

G. R. Khan, “Exact solution of N-dimensional radial Schrodinger equation for the fourth-order inverse-power potential,” The European Physical Journal D. Atomic, Molecular, Optical and Plasma Physics, vol. 53, no. 2, pp. 123–125, 2009.

S.-H. Dong, Z.-Q. Ma, and G. Esposito, “Exact solutions of the Schrödinger equation with inverse-power potential,” Foundations of Physics Letters, vol. 12, no. 5, pp. 465–474, 1999.

M. Hosseinpour, H. Hassanabadi, and M. Fabiano, “The DKP oscillator with a linear interaction in the cosmic string space-time,” Eur. Phys. J. C, vol. 78, no. 93, 2018.

B. D. B. Figueiredo, “Ince’s limits for confluent and double-confluent Heun equations,” Journal of Mathematical Physics, vol. 46, no. 11, Article ID 113503, 2005.

L. J. El-Jaick and B. D. Figueiredo, “Solutions for confluent and double-confluent Heun equations,” Journal of Mathematical Physics, vol. 49, no. 8, 083508, 28 pages, 2008.

S.-H. Dong, W.-C. Qiang, G.-H. Sun, and V. B. Bezerra, “Analytical approximations to the l-wave solutions of the Schrodinger equation with the Eckart potential,” Journal of Physics A: Mathematical and General, vol. 40, no. 34, pp. 10535–10540, 2007.

P. Malik, “Application of the exact quantization rule to the relativistic solution of the rotational Morse potential with pseudospin symmetry,” Journal of Physics A, vol. 40, pp. 1677–1685, 2007.

O. Bayrak and I. Boztosun, “The pseudospin symmetric solution of the Morse potential for any κ state,” Journal of Physics A: Mathematical and Theoretical, vol. 40, no. 36, pp. 11119–11127, 2007.

O. C. Aydoğdu and R. Sever, “Pseudospin and spin symmetry in the Dirac equation with Woods-Saxon potential and tensor potential,” The European Physical Journal A, vol. 43, article 73, 2010.

A. Soylu, O. Bayrak, and I. Boztosun, “An approximate solution of Dirac-Hulthén problem with pseudospin and spin symmetry for any κ state,” Journal of Mathematical Physics, vol. 48, Article ID 082302, 2007.

G. Chen, “Spinless particle in the generalized hulthén potential,” Modern Physics Letters A, vol. 19, no. 26, pp. 2009–2012, 2004.

M. Hosseinpoura, F. M. Andradeb, E. O. Silvac, and H. Hassanabadid, “Scattering and bound states of Dirac Equation in presence of cosmic string for Hulthén potential,” Eur. Phys. J. C, vol. 77, no. 270, 2017.

J. J. Pena, J. Morales, and J. Garcia-Ravelo, “Bound state solutions of Dirac equation with radial exponential-type potentials,” Journal of Mathematical Physics, vol. 58, Article ID 043501, 2017.

F. J. Ferreira and V. B. Bezerra, “Some remarks concerning the centrifugal term approximation,” Journal of Mathematical Physics, vol. 58, no. 10, 102104, 12 pages, 2017.

A. Tas, O. Aydogdu, and M. Salti, “Dirac particles interacting with the improved Frost-Musulin potential within the effective mass formalism,” Annals of Physics, vol. 379, pp. 67–82, 2017.

B. C. Lutfuğlu, F. Akdeniz, and O. Bayrak, “Scattering, bound, and quasi-bound states of the generalized symmetric Woods-Saxon potential,” Journal of Mathematical Physics, vol. 57, Article ID 032103, 2016.

H. Yukawa, “On the Interaction of Elementary Particles. I,” in Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, vol. 17, pp. 48–57, 1935.

A. Loeb and N. Weiner, “Cores in Dwarf Galaxies from Dark Matter with a Yukawa Potential,” Physical Review Letters, vol. 106, Article ID 171302, 2011.

F. Pakdel, A. A. Rajabi, and M. Hamzavi, “Scattering and Bound State Solutions of the Yukawa Potential within the Dirac Equation,” Advances in High Energy Physics, vol. 2014, Article ID 867483, 7 pages, 2014.

H. Bahlouli, M. S. Abdelmonem, and S. M. Al-Marzoug, “Analytical treatment of the oscillating Yukawa potential,” Chemical Physics, vol. 393, no. 1, pp. 153–156, 2012.

A. D. Alhaidari, H. Bahlouli, and M. S. Abdelmonem, “Taming the Yukawa potential singularity: improved evaluation of bound states and resonance energies,” Journal of Physics A: Mathematical and General, vol. 41, no. 3, 032001, 9 pages, 2008.

Altuğ Arda, “Solution of Effective-Mass Dirac Equation with Scalar-Vector and Pseudoscalar Terms for Generalized Hulthén Potential,” Advances in High Energy Physics, vol. 2017, Article ID 6340409, 9 pages, 2017.